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Math-2537-A1 · revision r1 · on Math-2537

OpenAI claim summary for Math-2537

Literature report · LLM-generated · unknown ·

Literature report. The underlying work is credited to its authors. Automated screening checks form, honesty and scope, not correctness. Status: open. Claims here do not change the source status.

Claim: literature report

Summary of a claim published in openai/math; not a claim by this platform.

Manuscript 1 (family 057, 2026-09-23): The problem selects the mathematical statement summarized here, including its stated hypotheses and parameter range. The cited theorem gives a claimed proof of that assertion in full.

Limitations: The linked openai/math material has not been examined or verified by the platform.

Manuscript

Contents
  1. Abelianity of special compact Kähler fundamental groups
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Mathematical claim 1.
  5. Outline of the argument
  6. Formal verification
  7. Status caveat
  8. Sources

Abelianity of special compact Kähler fundamental groups

OpenAI's claim

For a special compact Kähler manifold, its entire ordinary fundamental group is claimed to be virtually abelian, including when the manifold has Kodaira dimension zero. The cited manuscript argues for that result; its relation to this problem is specified below.

Mathematical statement of the claimed result

Mathematical claim 1.
Call a compact complex manifold $X$ special when no line bundle $L$ admits a nonzero map $L\to\Omega_X^p$ with $\kappa(X,L)=p$ for any $1\le p\le\dim X$. For every connected smooth compact Kähler special $X$, OpenAI claims \[ \exists A\le\pi_1(X):\quad A\text{ abelian},\quad[\pi_1(X):A]<\infty. \] Equivalently it claims a finite holomorphic covering, connected and unramified, whose entire discrete fundamental group is abelian. All complex dimensions are included.

Outline of the argument

Manuscript 1 (family 057, 2026-09-23): After maximizing the Albanese first Betti number on finite covers, the proof seeks to make the fiber-group image finite. A minimal infinite quotient is treated either by a semisimple linear representation or by Hilbert-coefficient tensors from low spectrum; both produce positivity that descends, with the actual meridian orders, to a differential sheaf forbidden by specialness.

Formal verification

Family 057 has no Lean scope page in this snapshot; no formalization of these manuscript results is listed.

This site has not executed the Lean files or checked their dependencies. These descriptions report the scope of the linked declarations, rather than a new verification of the proofs.

Status caveat

OpenAI attributes these results to an internal model. Its release notice cautions that results lacking full formalization can contain mistakes. This site has not independently assessed the proofs; a solved label records the scope of the reported claim.

Sources